We propose Reduced Collatz Conjecture that is equivalent to Collatz Conjecture, which states that every positive integer can return to an integer less than it, instead of 1. Reduced Collatz Conjecture should be easier because some properties are presented in reduced dynamics, rather than in original dynamics (e.g., ratio and period). Reduced dynamics is a computation sequence from starting integer to the first integer less than it, and original dynamics is a computation sequence from starting integer to 1. Reduced dynamics is a component of original dynamics. We denote dynamics of x as a sequence of either computations in terms of ``I'' that represents (3*x+1)/2 and ``O'' that represents x/2. Here 3*x+1 and x/2 are combined together, because 3*x+1 is always even and followed by x/2. We formally prove that all positive integers are partitioned into two halves and either presents ``I'' or ``O'' in next ongoing computation. More specifically, (1) if any positive integer x that is i module $2^t$ (i is an odd integer) is given, then the first t computations (each one is either ``I'' or ``O'' corresponding to whether current integer is odd or even) will be identical with that of i. (2) If current integer after t computations (in terms of ``I'' or ``O'') is less than x, then reduced dynamics of x is available. Otherwise, the residue class of x (namely, i module $2^t$) can be partitioned into two halves (namely, i module $2^{t+1}$ and $i+2^t$ module $2^{t+1}$), and either half presents ``I'' or ``O'' in intermediately forthcoming (t+1)-th computation.

Sensor fusion technology is one of extensive used methods in the field of robot, aerospace and target tracking control. In this paper, the generalized sensor fusion framework, named the closed-loop fusion (CLF) is analyzed and the optimal design principle of filter is proposed in detail. Fusion error optimization problem, which is the core issue of fusion design, is also solved better through the feedback compensation law of CLF framework. Differently from conventional methods, the fusion filter of CLF can be optimally designed and the determination of superposition of fusion information is avoided. To show the validity, simulation and experimental results are to be submitted.

In response to the increasing energy demand in Taiwan and the global trend of renewable energy development, Kuroshio energy is a potential energy source. How to extract this invaluable natural resource has then become an intriguing and important question in engineering practices. This study conducted a study for a nozzle-diffuser duct (NDD) as the Kuroshio currents energy harvester. The computational fluid dynamics (CFD) software ANSYS Fluent was employed to calculate the drag and added mass coefficients of the duct anchored to the seabed. Those coefficients were further imported into Orcaflex to simulate the motion of the duct under normal and storm wave conditions. Results showed that the duct was stable 25 m below the sea surface under normal wave condition. When the wave condition changed to storm waves, the duct needed to dive into at least 90 m below the sea surface to regain its stability and obtain high power take-off (PTO). An optimal design nozzle-diffuser-duct was reported and a PTO peak of 15 kW was expectable in the Kuroshio currents. Once a suitable offshore platform can be developed with sixty-six NDDs, a Megawatt Kuroshio ocean current power generation system is feasible in the near future.

Two-dimensional (2D) of transition metal dichalcogenides (TMDCs) are potential candidates for thermoelectric (TE) applications due to their unique structural properties. In this paper, we constructed an 2D monolayer TMDCs semiconductor γ-PbSn2(X=S, Se, Te) and first-principles calculations and Boltzmann transport theory are used to study the thermoelectric performance. We found that γ-PbSnX2 had an ultra-high carrier mobility up to 4.04×103 cm2V−1s−1 leading to a metal-like electrical conductivity. Meanwhile, γ-PbSnX2 both have high Seebeck coefficients, resulting in high power-factors, and also shows intrinsic low lattice thermal conductivity of 6-8 W/mK at room temperature. The lower lattice thermal conductivity and high power-factors resulted in excellent thermoelectric performance. The high ZT values of γ-PbSnS2 and γ-PbSnSe2 were as high as 2.65 and 2.96 at 900 K, respectively. The result suggests that the monolayer γ-PbSnX2 are better candidates for excellent thermoelectric performance.

Sharding shows great potential for extending the blockchain. The current challenge facing sharded blockchain technology lies in addressing the extended transaction confirmation times caused by isolated states between shards and unbalanced transaction allocation strategies. These factors contribute to an increase in cross-shard transactions and disproportionate shard workload, ultimately resulting in indefinite confirmation delays for cross-shard transactions. A critical priority for sharded blockchain systems is to conduct a comprehensive qualitative analysis to better understand and mitigate the prolonged transaction confirmation times. We introduce a Product-Form Queue Network (PFQN) model to address the transaction confirmation time problem in sharded blockchains and incorporate a new confirmation queue to more accurately simulate the actual transaction confirmation process in the blockchain. In addition, we provide a detailed quantitative analysis of the relationship between network load, consensus efficiency, and security in sharded blockchains, laying a theoretical foundation for achieving robustness and efficiency in sharded blockchains. This research not only contributes to addressing the scalability issues in sharded blockchains but also offers a new perspective for future research directions.

The Double U-pipe ground heat exchanger, recognized for its straightforward process, cost-effectiveness, high heat exchange efficiency, and low thermal resistance, stands as the predominant ground heat exchanger type in today's shallow geothermal energy development and utilization. Recent years have seen significant research into the heat transfer influencing factors and heat exchange performance of Double U-pipe ground heat ex-changers via experimental testing methods. Yet, studies integrating numerical simulation with in-situ testing have been less frequent. Drawing on the cylindrical heat source model theory and the outcomes of regional in-situ thermal response tests, this paper develops a Double U-pipe ground heat transfer model by establishing physical, mathematical, and heat transfer geometric models. It assesses the impact of varying inlet temperatures, flow rates, and initial ground temperatures on heat exchange efficiency under heating conditions. The results validate the precision of the double U-pipe ground heat exchanger model based on in-situ testing. They show that enhancing the temperature differential between the inlet and initial temperatures, elevating the initial ground temperature, and moderately increasing the flow rate can boost the system's heat exchange efficiency.